6.262.Lec14

# 6.262.Lec14 - DISCRETE STOCHASTIC PROCESSES Lecture 14...

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Lecture 14 - 3/19/2010 Discrete Stochastic Processes 1 DISCRETE STOCHASTIC PROCESSES Lecture 14 Section 3.6 Little’s Theorem Average waiting time of M/G/1 queue - the PK Formula “Pasta Property” - Poisson Arrivals See Time Averages Section 3.7 Mention “Delayed Renewal Processes”

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Lecture 14 - 3/19/2010 Discrete Stochastic Processes 2 Steady-State Renewal & Reward Theory Long-Term Time Averages Renewals Strong Law for Renewals () 1 lim 1 t Nt wp tX →∞ = ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ Rewards Strong Law for Renewal Rewards { } 0 1 lim t n t E R Rd ττ →∞ = Steady-State Expectations Elementary Renewal Theorem ( ) { } ( ) 1 lim lim tt ENt mt X →∞ →∞ = = Blackwell’s Theorem ( ) ( ) { } ( )( ) 1 lim lim X δ →∞ →∞ +− = = Arithmetic Case: ( ) ( ) 1 lim t mt nd mt nd X →∞ = Key Renewal Theorem Corollary 1 {} { } lim n t E R ERt X →∞ = Arithmetic Case: { } lim n k E R ERk d X →∞ = Steady-State Probabilities From Blackwell’s Theorem ( ) ( ) { } lim 1 t PNt →∞ + −= = ( ] { } lim exactly 1 renewal in , t Pt t →∞ + = o X Arithmetic Case: lim exactly 1 renewal at t d Pn d X →∞ = Steady-State Densities Assume X has a density () () ( ) , lim , , 0 X t Zt Xt fx f zx z x X →∞ = ≤< ( ) 1 lim lim X Yt Zt Fz fz f z X →∞ →∞ == ( ) lim X t Xt x X →∞ =
Lecture 14 - 3/19/2010 Discrete Stochastic Processes 3 () Lt = 0 1 [0, ] t L d t ττ = [0, ] [0, ] Wt A t t Little's Theorem A(t) D(t) 0S S 12 W 1 3 W A(t)= Arrival process D(t)=Departure process W 2 Renewal epochs number of customers in system (service + queue) at time t is a reward function = A(t) – D(t) Time averaged number of customers over [0,t] = n W = wait (in queue and service) for nth customer. (Neither independent nor identically distributed! Why?) Average wait for all customers arriving in [0,t] = For t in empty period before S 1 : i.e., (time averaged occupancy) = (average wait per customer) (average arrival rate) = * ( ) ( ) 11 123 0 ( ) [0, ] ( ) At kk t WW A t L d W tt t A t t == + + = = ∑∑ ( ) 1 [0, ] k k W A t = =

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Lecture 14 - 3/19/2010 Discrete Stochastic Processes 4 λ W .. 1 E[L ] ( ) n wp L X = This is also true for t in any empty period before S n , n = 1, 2, 3 , --- Little’s theorem asserts that this works in the limit of large t: Little’s Theorem (for G/G/1 Queue) = L
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## This note was uploaded on 10/21/2010 for the course EE 5581 taught by Professor Moon,j during the Spring '08 term at Minnesota.

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6.262.Lec14 - DISCRETE STOCHASTIC PROCESSES Lecture 14...

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